a path-integral approach to depth migration

A path-integral approach to depth migration Thomas A. Dickens* and Dennis E. Willen, ExxonMobil Upstream Research Company Summary Seismic depth migration can be cast as a weighted sum over forward paths connecting data measured on the surface to image locations in the subsurface. The integral forms of conventional downward-continuation operators have a sum-over-paths or path-integral interpretation. These path integrals are related to, but distinct from other path integrals that have appeared in the geophysical literature. Kirchhoff migration approximates the path sum by the contributions from a few, special raypaths. This connection to Kirchhoff migration may permit the computational domain to be trimmed sufficiently to apply path integrals to target-oriented, 3-D prestack depth migration. The effective use of path integrals for depth migration will require an appropriate Monte Carlo integration method, presumably developed around stationary or near-stationary points in the travel time field. Introduction The geophysical community has invested considerable effort and ingenuity in developing stable, efficient depth extrapolators for seismic data. For post-stack imaging, accuracy is an important issue at the large dips. For prestack imaging, it is valuable to remove the infinite-frequency and isolated-raypath restrictions of Kirchhoff methods. There are a variety of depth extrapolators available: the Dirac-equation method, the non-reflecting wave equation, the phase-screen method, explicit differential operators, and the classical, implicit methods both with and without error correction. While these techniques have been applied to 2-D and 3-D post-stack problems, they are impractical for 3-D prestack depth migration. Furthermore, these methods have only a formal, mathematical connection to Kirchhoff migration, making it difficult to develop new approximations around the familiar concepts of first-arrival and maximum-energy traveltimes. Integral propagators The exact, integral solution to the one-way wave equation (Fishman and McCoy, 1984) is often overlooked in these developments. The expressions for pressure are, in two dimensions,

)1()(

))((),,(

2),,(

220

220

)1(1

00∆+−

∆+−∆=∆+ ∫xx

xxHzxPdx

v

izxP v

ω

ωωω

and, in three dimensions,

[ ])2(.1

)()(

)()(

)()(),,,(

2),,,(

220

20

220

20

220

20

0000

−

∆+−+−

∆+−+−

∆+−+−

∆=∆+ ∫∫

yyxx

iv

yyxx

yyxxv

ixpezyxPdydx

v

izyxP

ω

ω

ωπωω

Here, v is the seismic velocity and H is a first-order Hankel function. Equations (1) and (2) can be incorporated into the usual migration formula based on summing over temporal frequencies. By way of example, Figure 1 shows a zero-offset time section over a simple, three-layered model. A high-velocity wedge, perhaps representing a salt sheet, has intruded into the middle layer from the right. The bottom reflector is pulled up in time below the wedge and there is a shadow zone beneath the wedge. The corresponding depth section is shown in Figure 2. The time data were downward continued layer by layer by applying Simpson’s rule to Equation (1). The bottom reflector has been restored to its proper position, the relative amplitudes of the reflections are apparent, and the shadow beneath the leading edge remains. It is also clear that the lower corner of the wedge has not been illuminated because data were not acquired sufficiently far to the right.

SEG 2000 Expanded Abstracts Main MenuSEG 2000 Expanded Abstracts Main Menu

Depth migration by path integrals

Figure 1. Zero-offset time section used for the migration test. A high-velocity wedge has been intruded into the

layers from the right.

Figure 2. Depth section obtained from the time section in Figure 1 by integrating equation (1). At zero offset, a

shadow zone is present on the lower reflector.

Downward continuation by path integral Equations (1) and (2) may be applied repeatedly to downward continue the wavefield to any depth desired. Defining

vxx jjj /)( 221 ∆+−= −ωα

and

vyyxx jjjjj /)()( 221

21 ∆+−+−= −−ωβ

we have

)3(),,,()(

2),,( 002

2)1(1

11 ωω

αα

ω zyxPv

Hdx

iNzxP

j

jN

jjN ∏ ∫

=−

∆=∆+

and,

)4().,,,(1)(

2),,,( 0023

3

111 ω

βββω

πω zyxP

i

v

ixpedydx

iNzyxP

jj

jN

jjjNN

−∆=∆+ ∏ ∫∫

=−−

Following prevailing practice, we have ignored gradients of the velocity and assumed that the spatial dependence of velocity can be restored when (3) and (4) are discretized A brief examination of (3) and (4) reveals that they are numerically equivalent to weighted sums over all forward paths

connecting 0x to x . Figure 3a depicts the explicit solution of (3) by a numerical integration rule. By simply rearranging the

terms, we get the integration rule depicted in Figure 3b. That is, the field is propagated over all forward paths connecting the initial position to the final position, with an oscillating weight factor. Thus, equations (3) and (4) qualify as path integrals, although they lack the elegant exponential form discovered by Feynman for the Schrodinger equation (Feynman, 1948).

SEG 2000 Expanded Abstracts Main MenuSEG 2000 Expanded Abstracts Main Menu

Depth migration by path integrals

The reasoning that leads to path integrals is somewhat subtle and worth restating. In a uniform earth, equations (1) and (2) are

valid for finite values of ∆ and tell us that the pressure field propagates along straight-line ray paths. Of course, in a uniform earth, straight rays are equivalent to first-arrival and maximum energy rays. In a non-uniform earth, (1) and (2) remain valid for

small enough ∆ and the reasoning in Figures 3a and 3b comes into play. That is, waves propagate along all possible paths connecting the initial and final locations, not just the classical ray path. In a uniform earth, the oscillatory integrands in (3) and

(4) must cancel each other for all paths except the classical ray path so that (1) and (2) can hold true for finite ∆ . Based on experience with first-arrival and maximum-energy Kirchhoff migrations, we expect (3) and (4) to be dominated by stationary or near-stationary paths corresponding to first-arrival rays, maximum-energy rays, reflected diffractions, etc.

Figure 3a. Portions of paths contributing to the brute-force solution of Equation (3). The pressure at each depth level is a function of the pressure throughout the previous depth.

Figure 3b. The terms in Equation (3) expressed as a path integral. The pressure at any point is a weighted sum over paths connecting that point to the pressure at the surface.

While the interpretation of a familiar formula as a path integral is elegant, it is only useful if effective approximations can be developed from the sum-over-paths concept. Bevc (1997) has already introduced one such approximation by implementing (3) as a semi-recursive Kirchhoff procedure. Bevc’s method iterates the sequence of traveltime computation, migration, and downward continuation to build the migrated image depth panel by depth panel. The results compare favorably to the more costly shot-profile migration by finite differences. By way of explanation, Bevc pointed out that his cascaded downward continuations effectively sample multiple ray paths between the surface and image locations (Figure 4) – an argument that is clearly correct in light of the path integral interpretation of (3) and (4).

Figure 4. Bevc's method of semi-recursive migration. Each downward continuation step involves a restricted range of paths but the migration effectively samples a large number of raypaths.

Other path integrals for the wave equation Path integrals have already appeared in the geophysical literature in work directed primarily at the forward modeling problem. Frazier (1987) based his frequency-domain path integral on the elastic versions of (1) and (2) and further introduced reflection and transmission coefficients to explicitly model seismic reflections from selected boundaries. His numerical method involves

SEG 2000 Expanded Abstracts Main MenuSEG 2000 Expanded Abstracts Main Menu

Depth migration by path integrals

direct integration over the reflecting and transmitting surfaces. Lomax (1999) posed his path integral in the time domain. His computational technique is based on Monte Carlo sampling of a subset of all possible rays. Emphasizing the two-way wave equation, Schlottmann (1999) has given an interesting time domain path integral involving multiple time derivatives of the source waveform. A number of authors, primarily in the ocean acoustics literature, have developed path integrals for the parabolic form of the wave equation by exploiting its similarity to the Schrodinger equation. Depth migration using path integrals Clearly, the downward continuation operators in (3) and (4) could be applied to either post-stack or prestack depth migration. As with any other wave-equation operators, we can expect these expressions to handle not just first- or maximum-energy arrivals, but all forward-propagating energy, including reflected diffractions. Relative to Kirchhoff migration, we could expect to improve the illumination of partial shadow zones and to convert some “noises” into useful signal. At first glance, the computational burden associated with path-integral migration would seem to preclude its use for 3-D prestack depth migration. Standard, finite-difference methods for shot-gather migration have a computational workload proportional to the volume of the image and to the migration aperture, which includes both the source-receiver distance and the volume of earth between the source and receivers to the image. Despite their huge computational cost, these methods enjoy two particular strengths. First, additional traces can be added to a shot gather with almost no increase in cost because the size of the migration aperture increases very slowly as consecutive receivers are added. In addition, since the entire measured wavefield is downard continued at once, most of the internal computations are done only once for all of the traces in the shot gather and for all of the image points. That is, the wavefield is only computed once at intermediate points between the surface and the image. By contrast, path-integral migration is more closely analogous to Kirchhoff migration, where every shot-receiver pair is downward continued individually to every point in the image. Like Kirchhoff methods, the work done in path-integral migration is multiplicative in the number of traces. Furthermore, since the integrands in (3) and (4) must be compounded along multiple paths connecting each trace to each image point, the wavefield at intermediate locations will be computed many times as they are crisscrossed by paths from different traces on the way to different image points. This extra effort disappears in Kirchhoff migration where the traveltime tables are rigged to transport the wavefield from surface to image in one, low-cost step. Therefore, path integrals will be an attractive integration method only under special circumstances. In particular, common-offset migrations must be performed on a trace-by-trace basis, dramatically increasing the cost of conventional shot-gather migrations and forcing the recalculation of the wavefield at intermediate points. The combination of common-offset and target-oriented 3-D migration further reduces the advantage of finite-difference methods because it is difficult to see how domain trimming can be implemented without an extra step of ray tracing. By making the integration paths explicit, path-integral methods may offer a more direct way to restrict the computational domain. The final hurdle to path-integral migration is the development of integration methods that will sample an “appropriate” distribution of paths. “Appropriate” means, of course, a distribution that will give the correct answer. Lomax (1999) recognizes the problem of obtaining a uniform distribution of path lengths, although that feature of the distribution may not be required. In the broader literature on path integrals, the integration is frequently carried out by a Monte Carlo process that seeks out the stationary points in the oscillating integrals – leading to first-arrival Fermat traveltimes in the geophysical problem. Presumably, much of the energy in (3) and (4) is carried along paths that are nearly stationary. Such paths could be found with standard, ray-shooting methods or by a Monte Carlo process that does not “stall” on stationary values of the traveltime. References Bevc, D., 1997, Imaging complex structures with semirecursive Kirchhoff migration, Geophysics, 62 (577-588). Feynman, R. P., 1948, Space-time approach to non-relativistic quantum mechanics, Reviews of Modern Physics, 20, 367-387. Fishman, L., and J. J. McCoy, 1984, Derivation of extended parabolic wave theories. II. Path integral representations, J. Math.

Phys, 25, 297-308. Frazier, N., 1987, Synthetic seismograms using multifold path integrals - I. Theory, Geophys. J. R. astr. Soc., 88, (621-646). Lomax, A., 1999, Path-summation waveforms, Geophys. J. Int., 138, (702-716) Schlottmann, R. B., 1999, A path integral formulation of acoustic wave propagation, Geophys. J. Int., 137, 353-363.

SEG 2000 Expanded Abstracts Main MenuSEG 2000 Expanded Abstracts Main Menu